This is the text of an email from Rick Perley on 23 December 1998. "Jim" is Jim Condon.
Jim and I see this issue from the same angle -- the confusion limit
is defined by the r.m.s. rumble in an image, caused by the response to
background sources in the beam, and from the uncleaned sidelobes of sources
outside the imaged area. However, this definition does not lead to a
clear prediction of the confusion limit, as the slope of the differential
number count is not steep enough to prevent the calculated answer being
set solely by the (single) strongest source in the sky. Even if we argue
that the strongest sources are all heavily resolved, so that the number
count will 'dive' at the upper end, we will still be left with the rms
confusion noise being set by the few strongest, unresolved, objects.
Failing a good, simple statistic, we fall back to real observations,
(which makes good sense to me!) The NVSS showed that the effective rms
noise added to snapshot images, presumably due to the multitude of
uncleaned background sources outside the primary beam, was about 300 microJy.
The limit will scale with beam solid angle, so we can estimate the effective
confusion for other configurations. And it will more or less scale with
mean spectral index, so we can scale to other bands.
Jim's measurement is appropriate for snapshots. For a long synthesis,
the confusion level should drop dramatically, as the sidelobe levels of
the synthesized beam decline. An estimate of this is wanting -- it's easy
to do with UVSIM or UVCON -- I'll do this when I get a few spare minutes.
For the sake of argument, let's assume that the rms sidelobe level
of the beam for a long synthesis is 1/10 of that for the NVSS snapshot.
I then get the following table for the rms confusion levels in microJy:
Band Configuration
--------------------------------------------
A B C D
--------------------------------------------
P (327) .5 5 50 500
L .03 .3 3 30
S .005 .05 .5 5
C .001 .01 .1 1
X .0004 .004 .04 .4
U really small .005 .05
K,Ka,Q not a problem!
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